How to project an image through the lens?
EEVEE is not supporting indirect lighting right now. You still can use Irradiance Volume and indirect light projection to Bake those effect into your scene.

If you set a proper light map, it might work

Refraction won't provide any light into the scene maybe. And it use some algorithm to achieve it in EEVEE, which is not focus on project light. In these kind of problem, we usually adopt ray tracing to mock the effect, which EEVEE is not good at it. EEVEE is rasterization rendering engine.